Integrand size = 33, antiderivative size = 172 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^3 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^4}+\frac {3 e (b d-a e)^2 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^4}+\frac {e^2 (b d-a e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{3 b^4}+\frac {e^3 (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{10 b^4} \]
1/7*(-a*e+b*d)^3*(b*x+a)^6*((b*x+a)^2)^(1/2)/b^4+3/8*e*(-a*e+b*d)^2*(b*x+a )^7*((b*x+a)^2)^(1/2)/b^4+1/3*e^2*(-a*e+b*d)*(b*x+a)^8*((b*x+a)^2)^(1/2)/b ^4+1/10*e^3*(b*x+a)^9*((b*x+a)^2)^(1/2)/b^4
Time = 1.07 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.71 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (210 a^6 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+252 a^5 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+210 a^4 b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+120 a^3 b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )+45 a^2 b^4 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+10 a b^5 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )+b^6 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right )}{840 (a+b x)} \]
(x*Sqrt[(a + b*x)^2]*(210*a^6*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 252*a^5*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 210*a^4*b ^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 120*a^3*b^3*x^3 *(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 45*a^2*b^4*x^4*(56*d^ 3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 10*a*b^5*x^5*(84*d^3 + 216 *d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + b^6*x^6*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 + 84*e^3*x^3)))/(840*(a + b*x))
Time = 0.40 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^6 (d+e x)^3dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^6 (d+e x)^3dx}{a+b x}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {e^3 (a+b x)^9}{b^3}+\frac {3 e^2 (b d-a e) (a+b x)^8}{b^3}+\frac {3 e (b d-a e)^2 (a+b x)^7}{b^3}+\frac {(b d-a e)^3 (a+b x)^6}{b^3}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {e^2 (a+b x)^9 (b d-a e)}{3 b^4}+\frac {3 e (a+b x)^8 (b d-a e)^2}{8 b^4}+\frac {(a+b x)^7 (b d-a e)^3}{7 b^4}+\frac {e^3 (a+b x)^{10}}{10 b^4}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(((b*d - a*e)^3*(a + b*x)^7)/(7*b^4) + (3*e *(b*d - a*e)^2*(a + b*x)^8)/(8*b^4) + (e^2*(b*d - a*e)*(a + b*x)^9)/(3*b^4 ) + (e^3*(a + b*x)^10)/(10*b^4)))/(a + b*x)
3.20.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(379\) vs. \(2(120)=240\).
Time = 0.44 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.21
| method | result | size |
| gosper | \(\frac {x \left (84 e^{3} b^{6} x^{9}+560 x^{8} a \,b^{5} e^{3}+280 x^{8} d \,e^{2} b^{6}+1575 x^{7} e^{3} b^{4} a^{2}+1890 x^{7} d \,e^{2} b^{5} a +315 x^{7} d^{2} e \,b^{6}+2400 x^{6} e^{3} a^{3} b^{3}+5400 x^{6} d \,e^{2} b^{4} a^{2}+2160 x^{6} d^{2} e \,b^{5} a +120 x^{6} d^{3} b^{6}+2100 x^{5} e^{3} a^{4} b^{2}+8400 x^{5} d \,e^{2} a^{3} b^{3}+6300 x^{5} d^{2} e \,b^{4} a^{2}+840 x^{5} d^{3} b^{5} a +1008 x^{4} e^{3} a^{5} b +7560 x^{4} d \,e^{2} a^{4} b^{2}+10080 x^{4} d^{2} e \,a^{3} b^{3}+2520 x^{4} d^{3} b^{4} a^{2}+210 x^{3} e^{3} a^{6}+3780 x^{3} d \,e^{2} a^{5} b +9450 x^{3} d^{2} e \,a^{4} b^{2}+4200 x^{3} d^{3} a^{3} b^{3}+840 a^{6} d \,e^{2} x^{2}+5040 a^{5} b \,d^{2} e \,x^{2}+4200 a^{4} b^{2} d^{3} x^{2}+1260 x \,d^{2} e \,a^{6}+2520 x \,d^{3} a^{5} b +840 d^{3} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 \left (b x +a \right )^{5}}\) | \(380\) |
| default | \(\frac {x \left (84 e^{3} b^{6} x^{9}+560 x^{8} a \,b^{5} e^{3}+280 x^{8} d \,e^{2} b^{6}+1575 x^{7} e^{3} b^{4} a^{2}+1890 x^{7} d \,e^{2} b^{5} a +315 x^{7} d^{2} e \,b^{6}+2400 x^{6} e^{3} a^{3} b^{3}+5400 x^{6} d \,e^{2} b^{4} a^{2}+2160 x^{6} d^{2} e \,b^{5} a +120 x^{6} d^{3} b^{6}+2100 x^{5} e^{3} a^{4} b^{2}+8400 x^{5} d \,e^{2} a^{3} b^{3}+6300 x^{5} d^{2} e \,b^{4} a^{2}+840 x^{5} d^{3} b^{5} a +1008 x^{4} e^{3} a^{5} b +7560 x^{4} d \,e^{2} a^{4} b^{2}+10080 x^{4} d^{2} e \,a^{3} b^{3}+2520 x^{4} d^{3} b^{4} a^{2}+210 x^{3} e^{3} a^{6}+3780 x^{3} d \,e^{2} a^{5} b +9450 x^{3} d^{2} e \,a^{4} b^{2}+4200 x^{3} d^{3} a^{3} b^{3}+840 a^{6} d \,e^{2} x^{2}+5040 a^{5} b \,d^{2} e \,x^{2}+4200 a^{4} b^{2} d^{3} x^{2}+1260 x \,d^{2} e \,a^{6}+2520 x \,d^{3} a^{5} b +840 d^{3} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{840 \left (b x +a \right )^{5}}\) | \(380\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{3} b^{6} x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a \,b^{5} e^{3}+3 d \,e^{2} b^{6}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 e^{3} b^{4} a^{2}+18 d \,e^{2} b^{5} a +3 d^{2} e \,b^{6}\right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (20 e^{3} a^{3} b^{3}+45 d \,e^{2} b^{4} a^{2}+18 d^{2} e \,b^{5} a +d^{3} b^{6}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 e^{3} a^{4} b^{2}+60 d \,e^{2} a^{3} b^{3}+45 d^{2} e \,b^{4} a^{2}+6 d^{3} b^{5} a \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 e^{3} a^{5} b +45 d \,e^{2} a^{4} b^{2}+60 d^{2} e \,a^{3} b^{3}+15 d^{3} b^{4} a^{2}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{3} a^{6}+18 d \,e^{2} a^{5} b +45 d^{2} e \,a^{4} b^{2}+20 d^{3} a^{3} b^{3}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 d \,e^{2} a^{6}+18 d^{2} e \,a^{5} b +15 d^{3} a^{4} b^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (3 d^{2} e \,a^{6}+6 d^{3} a^{5} b \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d^{3} a^{6} x}{b x +a}\) | \(493\) |
1/840*x*(84*b^6*e^3*x^9+560*a*b^5*e^3*x^8+280*b^6*d*e^2*x^8+1575*a^2*b^4*e ^3*x^7+1890*a*b^5*d*e^2*x^7+315*b^6*d^2*e*x^7+2400*a^3*b^3*e^3*x^6+5400*a^ 2*b^4*d*e^2*x^6+2160*a*b^5*d^2*e*x^6+120*b^6*d^3*x^6+2100*a^4*b^2*e^3*x^5+ 8400*a^3*b^3*d*e^2*x^5+6300*a^2*b^4*d^2*e*x^5+840*a*b^5*d^3*x^5+1008*a^5*b *e^3*x^4+7560*a^4*b^2*d*e^2*x^4+10080*a^3*b^3*d^2*e*x^4+2520*a^2*b^4*d^3*x ^4+210*a^6*e^3*x^3+3780*a^5*b*d*e^2*x^3+9450*a^4*b^2*d^2*e*x^3+4200*a^3*b^ 3*d^3*x^3+840*a^6*d*e^2*x^2+5040*a^5*b*d^2*e*x^2+4200*a^4*b^2*d^3*x^2+1260 *a^6*d^2*e*x+2520*a^5*b*d^3*x+840*a^6*d^3)*((b*x+a)^2)^(5/2)/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (120) = 240\).
Time = 0.72 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.90 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{10} \, b^{6} e^{3} x^{10} + a^{6} d^{3} x + \frac {1}{3} \, {\left (b^{6} d e^{2} + 2 \, a b^{5} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (b^{6} d^{2} e + 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{3} + 18 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} + 20 \, a^{3} b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{3} + 15 \, a^{2} b^{4} d^{2} e + 20 \, a^{3} b^{3} d e^{2} + 5 \, a^{4} b^{2} e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (5 \, a^{2} b^{4} d^{3} + 20 \, a^{3} b^{3} d^{2} e + 15 \, a^{4} b^{2} d e^{2} + 2 \, a^{5} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{3} + 45 \, a^{4} b^{2} d^{2} e + 18 \, a^{5} b d e^{2} + a^{6} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{3} + 6 \, a^{5} b d^{2} e + a^{6} d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (2 \, a^{5} b d^{3} + a^{6} d^{2} e\right )} x^{2} \]
1/10*b^6*e^3*x^10 + a^6*d^3*x + 1/3*(b^6*d*e^2 + 2*a*b^5*e^3)*x^9 + 3/8*(b ^6*d^2*e + 6*a*b^5*d*e^2 + 5*a^2*b^4*e^3)*x^8 + 1/7*(b^6*d^3 + 18*a*b^5*d^ 2*e + 45*a^2*b^4*d*e^2 + 20*a^3*b^3*e^3)*x^7 + 1/2*(2*a*b^5*d^3 + 15*a^2*b ^4*d^2*e + 20*a^3*b^3*d*e^2 + 5*a^4*b^2*e^3)*x^6 + 3/5*(5*a^2*b^4*d^3 + 20 *a^3*b^3*d^2*e + 15*a^4*b^2*d*e^2 + 2*a^5*b*e^3)*x^5 + 1/4*(20*a^3*b^3*d^3 + 45*a^4*b^2*d^2*e + 18*a^5*b*d*e^2 + a^6*e^3)*x^4 + (5*a^4*b^2*d^3 + 6*a ^5*b*d^2*e + a^6*d*e^2)*x^3 + 3/2*(2*a^5*b*d^3 + a^6*d^2*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 15698 vs. \(2 (121) = 242\).
Time = 1.45 (sec) , antiderivative size = 15698, normalized size of antiderivative = 91.27 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**5*e**3*x**9/10 + x**8*(51* a*b**6*e**3/10 + 3*b**7*d*e**2)/(9*b**2) + x**7*(201*a**2*b**5*e**3/10 + 2 1*a*b**6*d*e**2 - 17*a*(51*a*b**6*e**3/10 + 3*b**7*d*e**2)/(9*b) + 3*b**7* d**2*e)/(8*b**2) + x**6*(35*a**3*b**4*e**3 + 63*a**2*b**5*d*e**2 - 8*a**2* (51*a*b**6*e**3/10 + 3*b**7*d*e**2)/(9*b**2) + 21*a*b**6*d**2*e - 15*a*(20 1*a**2*b**5*e**3/10 + 21*a*b**6*d*e**2 - 17*a*(51*a*b**6*e**3/10 + 3*b**7* d*e**2)/(9*b) + 3*b**7*d**2*e)/(8*b) + b**7*d**3)/(7*b**2) + x**5*(35*a**4 *b**3*e**3 + 105*a**3*b**4*d*e**2 + 63*a**2*b**5*d**2*e - 7*a**2*(201*a**2 *b**5*e**3/10 + 21*a*b**6*d*e**2 - 17*a*(51*a*b**6*e**3/10 + 3*b**7*d*e**2 )/(9*b) + 3*b**7*d**2*e)/(8*b**2) + 7*a*b**6*d**3 - 13*a*(35*a**3*b**4*e** 3 + 63*a**2*b**5*d*e**2 - 8*a**2*(51*a*b**6*e**3/10 + 3*b**7*d*e**2)/(9*b* *2) + 21*a*b**6*d**2*e - 15*a*(201*a**2*b**5*e**3/10 + 21*a*b**6*d*e**2 - 17*a*(51*a*b**6*e**3/10 + 3*b**7*d*e**2)/(9*b) + 3*b**7*d**2*e)/(8*b) + b* *7*d**3)/(7*b))/(6*b**2) + x**4*(21*a**5*b**2*e**3 + 105*a**4*b**3*d*e**2 + 105*a**3*b**4*d**2*e + 21*a**2*b**5*d**3 - 6*a**2*(35*a**3*b**4*e**3 + 6 3*a**2*b**5*d*e**2 - 8*a**2*(51*a*b**6*e**3/10 + 3*b**7*d*e**2)/(9*b**2) + 21*a*b**6*d**2*e - 15*a*(201*a**2*b**5*e**3/10 + 21*a*b**6*d*e**2 - 17*a* (51*a*b**6*e**3/10 + 3*b**7*d*e**2)/(9*b) + 3*b**7*d**2*e)/(8*b) + b**7*d* *3)/(7*b**2) - 11*a*(35*a**4*b**3*e**3 + 105*a**3*b**4*d*e**2 + 63*a**2*b* *5*d**2*e - 7*a**2*(201*a**2*b**5*e**3/10 + 21*a*b**6*d*e**2 - 17*a*(51...
Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (120) = 240\).
Time = 0.21 (sec) , antiderivative size = 693, normalized size of antiderivative = 4.03 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e^{3} x^{3}}{10 \, b} + \frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d^{3} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4} e^{3} x}{6 \, b^{3}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a e^{3} x^{2}}{90 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} d^{3}}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{5} e^{3}}{6 \, b^{4}} + \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2} e^{3} x}{180 \, b^{3}} - \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{3} e^{3}}{1260 \, b^{4}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3} x}{6 \, b^{3}} + \frac {{\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} x}{2 \, b^{2}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a x}{6 \, b} + \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, b^{2}} - \frac {{\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{4}}{6 \, b^{4}} + \frac {{\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{3}}{2 \, b^{3}} - \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2}}{6 \, b^{2}} - \frac {11 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a x}{72 \, b^{3}} + \frac {3 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} x}{8 \, b^{2}} + \frac {83 \, {\left (3 \, b d e^{2} + a e^{3}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a^{2}}{504 \, b^{4}} - \frac {27 \, {\left (b d^{2} e + a d e^{2}\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} a}{56 \, b^{3}} + \frac {{\left (b d^{3} + 3 \, a d^{2} e\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, b^{2}} \]
1/10*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e^3*x^3/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*d^3*x + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4*e^3*x/b^3 - 1 3/90*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*e^3*x^2/b^2 + 1/6*(b^2*x^2 + 2*a*b* x + a^2)^(5/2)*a^2*d^3/b + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^5*e^3/b^4 + 29/180*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a^2*e^3*x/b^3 - 209/1260*(b^2*x^ 2 + 2*a*b*x + a^2)^(7/2)*a^3*e^3/b^4 - 1/6*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3*x/b^3 + 1/2*(b*d^2*e + a*d*e^2)*(b^2*x^2 + 2*a*b* x + a^2)^(5/2)*a^2*x/b^2 - 1/6*(b*d^3 + 3*a*d^2*e)*(b^2*x^2 + 2*a*b*x + a^ 2)^(5/2)*a*x/b + 1/9*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x ^2/b^2 - 1/6*(3*b*d*e^2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^4/b^4 + 1/2*(b*d^2*e + a*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^3/b^3 - 1/6*(b* d^3 + 3*a*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2/b^2 - 11/72*(3*b*d*e^ 2 + a*e^3)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a*x/b^3 + 3/8*(b*d^2*e + a*d*e^ 2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*x/b^2 + 83/504*(3*b*d*e^2 + a*e^3)*(b^2 *x^2 + 2*a*b*x + a^2)^(7/2)*a^2/b^4 - 27/56*(b*d^2*e + a*d*e^2)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*a/b^3 + 1/7*(b*d^3 + 3*a*d^2*e)*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (120) = 240\).
Time = 0.28 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.39 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{10} \, b^{6} e^{3} x^{10} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, b^{6} d e^{2} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, a b^{5} e^{3} x^{9} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{8} \, b^{6} d^{2} e x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{4} \, a b^{5} d e^{2} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{8} \, a^{2} b^{4} e^{3} x^{8} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {18}{7} \, a b^{5} d^{2} e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {45}{7} \, a^{2} b^{4} d e^{2} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{7} \, a^{3} b^{3} e^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a^{2} b^{4} d^{2} e x^{6} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} b^{3} d e^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b^{2} e^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{3} b^{3} d^{2} e x^{5} \mathrm {sgn}\left (b x + a\right ) + 9 \, a^{4} b^{2} d e^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, a^{5} b e^{3} x^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {45}{4} \, a^{4} b^{2} d^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {9}{2} \, a^{5} b d e^{2} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a^{6} e^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{5} b d^{2} e x^{3} \mathrm {sgn}\left (b x + a\right ) + a^{6} d e^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{6} d^{2} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{6} d^{3} x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (120 \, a^{7} b^{3} d^{3} - 45 \, a^{8} b^{2} d^{2} e + 10 \, a^{9} b d e^{2} - a^{10} e^{3}\right )} \mathrm {sgn}\left (b x + a\right )}{840 \, b^{4}} \]
1/10*b^6*e^3*x^10*sgn(b*x + a) + 1/3*b^6*d*e^2*x^9*sgn(b*x + a) + 2/3*a*b^ 5*e^3*x^9*sgn(b*x + a) + 3/8*b^6*d^2*e*x^8*sgn(b*x + a) + 9/4*a*b^5*d*e^2* x^8*sgn(b*x + a) + 15/8*a^2*b^4*e^3*x^8*sgn(b*x + a) + 1/7*b^6*d^3*x^7*sgn (b*x + a) + 18/7*a*b^5*d^2*e*x^7*sgn(b*x + a) + 45/7*a^2*b^4*d*e^2*x^7*sgn (b*x + a) + 20/7*a^3*b^3*e^3*x^7*sgn(b*x + a) + a*b^5*d^3*x^6*sgn(b*x + a) + 15/2*a^2*b^4*d^2*e*x^6*sgn(b*x + a) + 10*a^3*b^3*d*e^2*x^6*sgn(b*x + a) + 5/2*a^4*b^2*e^3*x^6*sgn(b*x + a) + 3*a^2*b^4*d^3*x^5*sgn(b*x + a) + 12* a^3*b^3*d^2*e*x^5*sgn(b*x + a) + 9*a^4*b^2*d*e^2*x^5*sgn(b*x + a) + 6/5*a^ 5*b*e^3*x^5*sgn(b*x + a) + 5*a^3*b^3*d^3*x^4*sgn(b*x + a) + 45/4*a^4*b^2*d ^2*e*x^4*sgn(b*x + a) + 9/2*a^5*b*d*e^2*x^4*sgn(b*x + a) + 1/4*a^6*e^3*x^4 *sgn(b*x + a) + 5*a^4*b^2*d^3*x^3*sgn(b*x + a) + 6*a^5*b*d^2*e*x^3*sgn(b*x + a) + a^6*d*e^2*x^3*sgn(b*x + a) + 3*a^5*b*d^3*x^2*sgn(b*x + a) + 3/2*a^ 6*d^2*e*x^2*sgn(b*x + a) + a^6*d^3*x*sgn(b*x + a) + 1/840*(120*a^7*b^3*d^3 - 45*a^8*b^2*d^2*e + 10*a^9*b*d*e^2 - a^10*e^3)*sgn(b*x + a)/b^4
Timed out. \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]